A Calabi-Yau pair of complexity two is cluster type exactly when its standard model over a toric base is nodal, component-compatible, and volume-large; this classifies all rank-one Gorenstein del Pezzo surfaces.
Log Calabi--Yau pairs of complexity zero and arbitrary index
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abstract
In this article, we give a characterization of log Calabi--Yau pairs of complexity zero and arbitrary index. As an application, we show that a log Calabi--Yau pair of birational complexity zero admits a crepant birational model which is a generalized Bott tower.
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Calabi-Yau pairs of complexity two
A Calabi-Yau pair of complexity two is cluster type exactly when its standard model over a toric base is nodal, component-compatible, and volume-large; this classifies all rank-one Gorenstein del Pezzo surfaces.